Question Number 137048 by mnjuly1970 last updated on 29/Mar/21
$$\:\:\:\:\:\:\:\:\:\:\:……..{advanced}\:\:…\:…\:…\:{calculus}……. \\ $$$$\:\:\:\:{evaluate}:::: \\ $$$$\:\:\:\:\:\:\boldsymbol{\chi}=\int_{\mathrm{0}} ^{\:\infty} {x}^{\mathrm{2}} \:{e}^{β{x}^{\mathrm{2}} } {ln}\left({x}\right)=??? \\ $$$$\:\: \\ $$
Answered by Ar Brandon last updated on 29/Mar/21
$$\boldsymbol{\chi}=\int_{\mathrm{0}} ^{\infty} \mathrm{x}^{\mathrm{2}} \mathrm{e}^{β\mathrm{x}^{\mathrm{2}} } \mathrm{lnxdx}=\frac{\partial}{\partial\mathrm{t}}\mid_{\mathrm{t}=\mathrm{0}} \int_{\mathrm{0}} ^{\infty} \mathrm{x}^{\mathrm{2}+\mathrm{t}} \mathrm{e}^{β\mathrm{x}^{\mathrm{2}} } \mathrm{dx} \\ $$$$\overset{\left(\mathrm{u}=\mathrm{x}^{\mathrm{2}} \right)} {=}\frac{\partial}{\partial\mathrm{t}}\mid_{\mathrm{t}=\mathrm{0}} \frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} \mathrm{u}^{\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{t}}{\mathrm{2}}} \mathrm{e}^{β\mathrm{u}} \mathrm{du}=\frac{\partial}{\partial\mathrm{t}}\mid_{\mathrm{t}=\mathrm{0}} \frac{\mathrm{1}}{\mathrm{2}}\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}+\frac{\mathrm{t}}{\mathrm{2}}\right) \\ $$$$\:\:\:=\frac{\mathrm{1}}{\mathrm{2}}\mid_{\mathrm{t}=\mathrm{0}} \frac{\mathrm{1}}{\mathrm{2}}\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}+\frac{\mathrm{t}}{\mathrm{2}}\right)\psi\left(\frac{\mathrm{3}}{\mathrm{2}}+\frac{\mathrm{t}}{\mathrm{2}}\right) \\ $$$$\:\:\:=\frac{\mathrm{1}}{\mathrm{4}}\Gamma\left(\frac{\mathrm{3}}{\mathrm{2}}\right)\psi\left(\frac{\mathrm{3}}{\mathrm{2}}\right)=\frac{\mathrm{1}}{\mathrm{4}}\centerdot\frac{\sqrt{\pi}}{\mathrm{2}}\centerdot\left(\mathrm{2}β\gammaβ\mathrm{2ln2}\right) \\ $$
Commented by mnjuly1970 last updated on 29/Mar/21
$$\:{very}\:{very}\:{nice} \\ $$$$\:\:{mr}\:{brandon}\:..{thank}\:{you}… \\ $$
Commented by Ar Brandon last updated on 29/Mar/21
My pleasure, Sir
Answered by Dwaipayan Shikari last updated on 29/Mar/21
$$\int_{\mathrm{0}} ^{\infty} {x}^{\mathrm{2}} {e}^{β{x}^{\mathrm{2}} } {log}\left({x}\right){dx}=\aleph'\left(\mathrm{3}\right) \\ $$$$\aleph\left(\alpha\right)=\int_{\mathrm{0}} ^{\infty} {x}^{\alphaβ\mathrm{1}} {e}^{β{x}^{\mathrm{2}} } {dx}\Rightarrow\aleph\left(\alpha\right)=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} {j}^{\frac{\alpha}{\mathrm{2}}β\mathrm{1}} {e}^{β{j}} {dj} \\ $$$$\aleph\left(\alpha\right)=\frac{\Gamma\left(\frac{\alpha}{\mathrm{2}}\right)}{\mathrm{2}}\Rightarrow\aleph'\left(\alpha\right)=\frac{\Gamma'\left(\frac{\alpha}{\mathrm{2}}\right)}{\mathrm{4}}\Rightarrow\aleph'\left(\mathrm{3}\right)=\frac{\sqrt{\pi}}{\mathrm{8}}\left(β\gamma+\mathrm{2}β\mathrm{2}{log}\left(\mathrm{2}\right)\right) \\ $$
Commented by mnjuly1970 last updated on 29/Mar/21
$${thanks}\:{alot}\:{mr}\:{payan}… \\ $$
Commented by Dwaipayan Shikari last updated on 29/Mar/21
Have a great day sir!
Answered by mathmax by abdo last updated on 30/Mar/21
$$\Phi=\int_{\mathrm{0}} ^{\infty} \:\mathrm{x}^{\mathrm{2}} \:\mathrm{e}^{β\mathrm{x}^{\mathrm{2}} } \mathrm{lnx}\:\mathrm{dx}\:\:\mathrm{let}\:\mathrm{f}\left(\mathrm{a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\mathrm{x}^{\mathrm{a}} \:\mathrm{e}^{β\mathrm{x}^{\mathrm{2}} } \mathrm{dx} \\ $$$$\mathrm{v}\left(\mathrm{o}\right)\:\:\mathrm{x}^{\mathrm{a}} \:\mathrm{e}^{β\mathrm{x}^{\mathrm{2}} } \sim\mathrm{x}^{\mathrm{a}} \:\:\:\:\int_{\mathrm{0}} ^{\xi} \:\:\frac{\mathrm{dx}}{\mathrm{x}^{β\mathrm{a}} }\:\mathrm{cv}\:\Leftrightarrowβ\mathrm{a}<\mathrm{1}\:\Rightarrow\mathrm{a}>β\mathrm{1} \\ $$$$\mathrm{we}\:\mathrm{have}\:\mathrm{f}\left(\mathrm{a}\right)=\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{\mathrm{alogx}} \:\mathrm{e}^{β\mathrm{x}^{\mathrm{2}} } \mathrm{dx}\:\Rightarrow\mathrm{f}^{'} \left(\mathrm{a}\right)=\int_{\mathrm{0}} ^{\infty} \mathrm{x}^{\mathrm{a}} \:\mathrm{e}^{β\mathrm{x}^{\mathrm{2}} } \mathrm{logx}\:\mathrm{dx} \\ $$$$\mathrm{and}\:\mathrm{f}^{'} \left(\mathrm{2}\right)=\int_{\mathrm{0}} ^{\infty} \:\mathrm{x}^{\mathrm{2}} \:\mathrm{e}^{β\mathrm{x}^{\mathrm{2}} } \mathrm{logx}\:\mathrm{dx} \\ $$$$\mathrm{changement}\:\mathrm{x}^{\mathrm{2}} =\mathrm{t}\:\mathrm{givef}\left(\mathrm{a}\right)=\int_{\mathrm{0}} ^{\infty} \:\mathrm{t}^{\frac{\mathrm{a}}{\mathrm{2}}} \:\mathrm{e}^{β\mathrm{t}} \:\:\frac{\mathrm{dt}}{\mathrm{2}\sqrt{\mathrm{t}}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} \:\mathrm{t}^{\frac{\mathrm{a}β\mathrm{1}}{\mathrm{2}}} \:\mathrm{e}^{β\mathrm{t}} \:\mathrm{dt}\:=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} \:\mathrm{t}^{\frac{\mathrm{a}β\mathrm{1}}{\mathrm{2}}+\mathrm{1}β\mathrm{1}} \:\mathrm{e}^{β\mathrm{t}} \mathrm{dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty} \:\mathrm{t}^{\frac{\mathrm{a}+\mathrm{1}}{\mathrm{2}}β\mathrm{1}} \:\mathrm{e}^{β\mathrm{t}} \:\mathrm{dt}\:=\frac{\mathrm{1}}{\mathrm{2}}\Gamma\left(\frac{\mathrm{a}+\mathrm{1}}{\mathrm{2}}\right)\:\Rightarrow\mathrm{f}^{'} \left(\mathrm{a}\right)=\frac{\mathrm{1}}{\mathrm{4}}\Gamma^{'} \left(\frac{\mathrm{a}+\mathrm{1}}{\mathrm{2}}\right)\:\Rightarrow \\ $$$$\mathrm{f}^{'} \left(\mathrm{2}\right)\:=\frac{\mathrm{1}}{\mathrm{4}}\Gamma^{'} \left(\frac{\mathrm{3}}{\mathrm{2}}\right)\:=\Phi \\ $$$$ \\ $$